how to update environment variable in variable group so new value persists, so new value can be used even after build is finished.
I am afraid there is no such way to update environment variable in variable group and keep it persists after build is finished.
When you use the Logging Command to set the variable, which is environment variable and can only work in the current environment.
So the new value can not be used after build is finished.
on the other hand, just like Daniel said, if we write any persistent value, then this value will compete/conflict with the value in the variable group. The compiler will not know which value to choose.
So, if you want to write any persistent value, we have to update the value in the variable group manually or use REST API to update it in the variable group.
Check the How to modify Azure DevOps release definition variable from a release task? for some more details.
Hope this helps.